Surfaces close to the Severi lines

Abstract

In this Thesis we study surfaces of general type with maximal Albanese dimension for which the quantity $K_X^2-4χ(\mathcal{O}_X)-4(q-2)$ vanishes or is "small", that is surfaces close to the Severi lines. Over the complex numbers, it is known that a surface $X$, provided that $K_X^2<\frac{9}{2}χ(\mathcal{O}_X)$, has to satisfy the inequality $K_X^2-4χ(\mathcal{O}_X)-4(q-2)\geq 0$. We give a constructive and complete classification of surfaces for which equality holds: these are surfaces whose canonical model is a double cover of an Abelian surface ($q=2$) or of a product elliptic surface ($q\geq 3$) branched over an ample divisor with at most negligible singularities which intersects the elliptic fibre twice in the latter case. We also prove, in the same hypothesis, that a surface $X$ with $K_X^2\neq 4χ(\mathcal{O}_X)+4(q-2)$ satisfies $K_X^2\geq 4χ(\mathcal{O}_X)+8(q-2)$ and we give a characterization of surfaces for which the equality holds. These are surfaces whose canonical model is a double cover of an isotrivial smooth elliptic surface fibration branched over an ample divisor with at most negligible singularities whose intersection with the elliptic fibre is $4$. Because these results are intimately related to theory of double covers, we see that their proof extend almost step by step to the case of any algebraically closed field of characteristic different from $2$. We also give some partial results over algebraically closed fields of characteristic $2$ after a study of double covers in that case.